1. Field of the Invention
The present invention relates to filtering techniques for a system comprising a plurality of signals. More particularly, the invention relates to a finite impulse response (“FIR”) filter that equalizes group delay and magnitude in the system.
2. Description of the Prior Art
It is often preferable that a filter for a system comprising a plurality of signals is operable to equalize the group delay and magnitude of the system. Historically, infinite impulse response (“IIR”) filters have been more commonly used for equalization of the group delay and magnitude than finite impulse response (“FIR”) filters. This is because it is more difficult to design an FIR filter to have a non-linear phase than it is for IIR filters. Additionally, IIR filters have been chosen to equalize group delay and magnitude because of the perception that resulting FIR filter lengths will be excessively long.
Although IIR filters have historically been used to equalize group delay and magnitude, a limited number of prior art FIR filters have been designed to equalize group delay and magnitude. FIR filters may be preferred over IIR filters for equalization because FIR filters can be readily implemented in high-speed, low-cost digital hardware. If the equalization is implemented in software, FIR filters are generally simpler to implement because there is not a concern about accumulated round-off and stability problems. Thus, for many applications, FIR filters are preferred over IIR filters.
The typical FIR optimization or approximation technique is to solve for an equalizing filter, H(Ω), that minimizes a weighted error, E(Ω), between a desired response, D(Ω), and a cascaded response:E(Ω)=[D(Ω)−F(Ω)H(eiΩ)]W(Ω)The desire is to minimize the weighted error across the equalizing FIR filter tap weights:
      min          h      ⁡              (        k        )              ⁢                E      ⁡              (        Ω        )                where D(Ω) is the desired response as a function of frequency, F(Ω) is the given or known response to be equalized, h(k) is the equalizing FIR filter tap weights, and W(Ω) is the weighting.
As can be understood by one with ordinary skill in the art, this optimization technique optimizes only across the filter tap weights. Therefore, the minimization is performed on both the linear and non-linear components of a phase of the system. As such, the slope of the phase is arbitrary for each application of the optimization. This results in an equalization of the group delay and magnitude that is generally poor.
Additionally, because the system response is known, and therefore the phase response is known, and further because the optimization does not fix any component of the phase, the prior art FIR filter described above matches the desired phase response with the known phase response. Unfortunately, this matching is not consistently successful and is too dependent on the occurring phase slope.
Accordingly, there is a need for an improved filter that better equalizes group delay and magnitude. Specifically, there is a need for an improved filter design that fixes the phase slope as a parameter in the optimization or minimization, thus minimizing only the non-linear portion of the phase.